### Comparing Greenhouse Gases for Policy Purposes*

Richard Schmalensee**

In order to derive optimal policiesfor greenhouse gas emtsswns control, thediscountedmarginaldamages of emissions from differentgasesmust be compared. 1he greenhouse warmingpotential (GWP) index, which is most often used to compare greenhouse gases, is not based on such a damage comparison. 1his essaypresents assumptions underwhich ratios of gas-specific discounted marginal damages reduce to ratios of discounted marginal contributions to radiativeforcing, where the discount rate is the difference between the discount rate relevant to climate-related damages and the rate of growth of marginal climate-related damages over time. 

## Ifthere are important gas-specific costs or benefits not tied to radiativeforcing, however, such as direct effects of carbon dioxide onplantgrowth, there is in general no shortcut around explicit comparison of discounted net marginal damages.



### INTRODUCTION
David Wood understood early on both the high economic stakes involved in debates about global climate change and the potentially huge contribution that careful and objective economic analysis could make to those debates. While I was serving on the Council of Economic Advisers and concerned with climate change policy, David was actively and effectively building interest in climate change among energy economists at MIT and elsewhere. He and I

**~lked** 

### about the economics ofglobal change several times

The Energy Journal, Volume 14, Number 1. All Rights Reserved.
*I began work on this paper while a member ofthe Council ofEconomic Advisers and (finally) completed it with support from the MIT Center for Energy and Environmental Policy Research. I am indebted to Richard Eckaus and John Reilly for very useful discussions.
**Center for Energy and Environmental Policy Research, Massachusetts Institute of Technology, Cambridge, MA 02139.



### 245

### 246 I The Energy Journal 

### during 1990 

### and the early spring of 1991. 

### We agreed about many things, includingthe ongoing fusion ofenergy and environmental policies that this issue exemplifies, but we argued about the subject ofthis paper. I like to think that David would have found my ideas more persuasive in their present form.



### THE PROBLEM
In 

### its first report, Working Group I ofthe Intergovernmental Panel on Climate Change (the IPCC) observed that
The earth's climate is dependent upon the radiative balance of the atmosphere, which inturn depends upon the inputofsolar radiation and the atmospheric abundances of radiatively active trace gases (i.e., greenhouse gases), clouds and aerosols. (IPCC, 1990, 

### p. 7)
Global anthropogenic emissions ofthe various greenhouse gases are not in fixed proportions.• Thus, for instance, the ratio oftotal methane

**(C~)** 

### emissions to total carbon dioxide (CO:J emissions caused by human activity can be affected by a variety ofgovernmental and intergovernmental policies. It

### follows that the design ofan efficient global policy aimed at slowing the rate ofclimate change would necessarily involve decisions on how much, over time, to spend on the margin to reduce CH4 

### emissions and how much to spend to 

### reduce C02 

### emissions. This choice in turn must logically reflect the marginal damages associated with emissions ofeach kilogram ofCH4 

### and C02• 

### As the intensity, as well as the substance, of the debate on global climate change makes clear, future damages attributable to current greenhouse gas emissions are highly uncertain. This reflects uncertainty about at least (a) how changes in today's emissions would affect future atmospheric abundances of greenhouse gases and thus future radiative forcing, (b) 

### how changes in the time-path ofradiative forcing would affect future climates, and (c) how changes in future climates would affect some appropriate measure of human welfare. Perhaps because the second and third of these sources of uncertainty have seemed particularly important and intractable and because damage analysis involves difficult problems of economic valuation, a number of authors have proposed schemes for comparing the relative values of reducing emissions of greenhouse gases that reflect only atmospheric abundances and their radiative forcing implications.

1. Hereand in what follows "greenhousegases" shouldbeunderstoodto includeboth radiatively active gases and aerosols as well as othergases and aerosolsthat affect the formation or destruction of such gases and aerosols.

### Comparing Greenhouse Gases I 247


### This essay argues that comparisons among greenhouse gases that are useful for analysis of abatement policies cannot be made without significant economic input; the physical sciences cannot supply all necessary information. If 

### greenhouse gas comparisons are to inform policy design, they must be ultimately based on analysis ofmarginal costs and benefits. As the next section demonstrates, it follows that comparisons which begin and end with summaries of contributions to radiative forcing over time have no welfare-economic or policy-analytic justification. The section that follows then shows that under certain assumptions, comparisons of discounted marginal damages reduce to comparisons of appropriately discounted marginal effects on global radiative forcing. Thewelfare-economic foundations developed therecan supportrigorous evaluation ofalternative discount rates. The final section considers the implications of relaxing two key assumptions. First, if changing concentrations of individual greenhouse gases produce importantcosts orbenefitsthat are unrelated to global radiative forcing, explicit calculations of discounted net marginal damages cannot in general be avoided. In particular, ifthe effects ofchanges in atmospheric concentrations of C02 

### on plantgrowth are economically important, itwill generally be impossible to make policy-relevant comparisons between C02 

### and other greenhouse gases without explicit computation ofgas-specific marginal discounted net damages, including those associated with C02 

### fertilization. Second, following essentially all the relevant literature, the formal analysis that follows does not explicitly consider uncertainty. Some ofthe issues that would be encountered in doing so are also discussed in the last section ofthis paper.



### STARTING WITH RADIATIVE FORCING
Because none of the relevant physical, chemical, or economic relationships can be guaranteed to be linear, and some are clearly nonlinear, the analysis here focuses on derivatives with respect to gas-specific emissions evaluated along some baseline economic/environmental trajectory. Let R(r) 

### be instantaneous radiative forcing (the net radiative flux change at the tropopause, usually measured in watts per square meter) at timer, let E;(O) 

### be emissions of gas i 

### (usually measured in kilograms) at time 0, 

### perhaps the present, fori 

## = 1,

, N, 

### and let

T

**~** 

### 0, 

### i = 

### 1,

... ' N. (1)



### For any 

### i and T, a;(r) 

### depends on the instantaneous radiative forcing associated with increases in the atmospheric concentration of gas 

### i, on the dynamics of removal of gas 

### i from the atmosphere, and on the impact of increases in the248 I The Energy Journal


### concentration ofgas i on the concentration over time ofother greenhouse gases and their precursors. These functions are often thoughtofas inputs to economic analysis, butthis is incorrect. Themarginal radiative forcing effect ofincreasing , the concentration of any one gas depends on initial concentrations of that gas (because ofsaturation ofabsorptionbands) and on initial concentrations ofother gases (because ofchemical interactions and overlap ofabsorption bands). Since these initial concentrations at any time depend on earlier emissions, the a;(r) Junctions inevitably depend to some extent on an explicit or implicit long-run economic forecast. 

### Italso follows that the 

### a1

### functions may depend importantly
on what date is taken as time zero and on the impact of any large-scale emissions reductions policies. The most frequently-cited approach to comparing greenhouse gasses is the index of global warming potential (GWP) presented by Lashof and Ahuja (1990) and by the IPCC (1990, 1992):

i = 

### 1,

... , N, (2)



### where T

### must be specified and, by convention, gas 1 is C02• 

### Thus GWP1 

## = 1, and the idea is that ifGWP2 = 

### 2, for instance, then one can argue that reducing emissions of gas 2 is twice as valuable, kilogram for kilogram, as reducing emissions ofC02• 

### The most obvious problem with the GWP 

### measure is that the horizon, T, 

### is completely arbitrary. The choice ofhorizon can beimportantinthis setting because the atmospheric lifetimes (half-lives) of the various greenhouse gases differ substantially. The second IPCC report (1992, p. 56), for instance, lists estimated lifetimes thatvary from "days" (for NOx) to 

## ">500" years (for CFC14 and CFC-116). The first IPCC report (1990, p. 60) dealt with the lack of any welldefined, defensible procedure for choosing T

### by showing values of a set of GWP 1 

### for T = 

### 20, 100, and 500. 2 

### The corresponding GWPs 

### for HCFC-123, which has an estimated lifetime much shorter than that ofC02, 

### are 310, 85, and 29. Considering only major greenhouse gases, the corresponding GWPs 

### shown

2. The second IPCC report (1992) acknowledged the severity ofuncertainties attached to the "indirect effects" of emissions of several greenhouse gases and did not present "total" GWPs comparableto those in the first report.

**l'~**.-Comparing Greenhouse Gases I 249
for methane (CH.J are 63, 21, and 9. 3 Since 20 years is clearly too short an horizon for this problem, while 500 years seems an awfully long time in any context, the reader's attention is naturally drawn to 

### T = 100 as some sort of. reasonable compromise. This is an old trick: drafters ofdecision memoranda in the White House learn quickly to fight to have their preferred choice in the middle ofthe list ofoptionspresented. Any ofa wide range ofvalues of Tcan be made to appear a reasonable compromise in this fashion. Moreover, some have argued that extreme values ofTshould be used instead ofcompromise values. Hammond, Rodenburg, and Moomaw (1990, p. 705) advocate treating oneunit ofgas 

### i at time zero as equivalent to a;(O)IalO) units of C02• They argue that using only very short-run changes in radiative forcing serves to tie "observable current results directly to policy actions

...." At the other extreme, Smith and Ahuja (1990) seem to argue that it is most appropriate to consider the total effects over time ofcurrent emissions, and this seems to imply setting T

## = oo. Nordhaus (1991) presents only this measure in his recent survey. Lashof and Ahuja (1990, p. 531) note that "current radiative forcing may be considered more important from a policy viewpoint than radiative forcingoccurringinthe distantfuture

..." Accordingly, they considerdiscounted GWP (DGWP, say) measures based on discounted integrals of the a;(r) functions:

DGWP, 

# Ia;(r) e-"dr
0 

### "" i = 1,

... , N, (3)

where r is a discount rate that must be specified. Nordhaus (1990) appears to have arrived independently at this same approach. The use ofa fixed horizon as in (2), with everything occurring before Ttreated identicallyand everything after Tignored, has no supportin economic theory. Because of this, T is arbitrary in a fundamental sense: there are no sound economic arguments that could be used to fix its value. On the other hand, discounting cash flows- either actual cash flows orvalues ofcosts and benefits ofvarious sorts- orutilityflows overan infinite horizon is commonplace and easilyjustified. Thus the main problem with (3) is

3. The second IPCC report (1992, p. 56) indicatesthat these numbers incorporate the effects of a typographicalerror. Thecorrespondingnumbersgiventhere for the directeffects only ofCH4 are 35, 11, and 4.

### 250 I The Energy Journal


### different: it applies discounting to a physical quantity, incremental radiative forcing, that is only by coincidence proportional to either an actual orpotential stream ofcash orutilityovertime. As Eckaus (1992, p. 27) puts it, "Addingup physical measures of radiative forcing in different periods resulting from emissions at different times and places is, in an economic policy sense, like adding apples and oranges: it cannot be done." Because discounting has no economic rationale here, there is no way to apply economic analysis to the determination ofan appropriate value ofr. 

### Thus, though (3) has a more familiar and defensibleweightingthan (2), it is in the end equally arbitrary because there is no principled way to fix r.



### ENDING WITH RADIATIVE FORCING
This section considers comparisons based on the relation between changes in quantities ofemissions at time zero, the E;(O), 

### and D(t), 

### the dollar value of damages at time t 

### caused by climate change. As above, the analysis deals with derivatives along some baseline economic/environmental trajectory; here the focus is on derivatives ofdiscounteddamages with respect to emissions. Nordhaus (1991), Reilly (1991), Uzawa (1991), Eckaus (1992), and others have analyzed the first-order conditions for optimality of such a trajectory.4 

### While the approach taken here is formally somewhat more general because itconsiders derivatives along trajectories that may not be optimal, it should be clear that the
same gas-specific derivatives are central to both approaches. Suppose that the only external effects ofgreenhouse gas emissions are
on the global climate and that the state ofthe global climate can be adequately summarized for cost-benefit purposes by 

### M variables, where 

### M is finite. Let

**~(t)** be 

### the value of the/' of these climate variables at time t. 

### The most discussed such variable is global mean temperature, but other quantities, such
as regional values of soil moisture and tropical storm frequencies may be considerably more important. We use the following notation for the marginal relations between climate and damages (conditional on whatever assumption regarding adaptation seems most reasonable):

4. Eckaus (1992) considers cost minimization subject to an exogenously-imposedtime-path of mdiative forcing, while the others treat mdiative forcing as determined by the optimization. While the resulting formulae differ somewhat in form, there is no substantive difference relevant to the issues considered here.Comparing Greenhouse Gases I 251
fJD(t)lfJS(u) 

# = f3p,u), t

**~** u, j = 

### 1,

... , M. (4)



### Since climate variables depend on the historical time-path of radiative forcing (and, perhaps, other quantities that will be assumed to be 

### exogenous), we can define fJS(u)!fJR(r) 

# = 'Y}Il,T), IJ

**~** T, j = 

### 1,

... ' M. (5)


### Using (4) and (5), the marginal relation between radiative forcing and climate-related damages becomes:

fJD(t)lfJR(r) 

## =

# E[I"' [fJD(t)lfJCiu)][fJS(u)tfJR(r)Jdu] }•I 0 (6)



## =

# t [I"' f3p,u)'Yiu,r)du] 

# = q,(t,r).

J•l 0


### Finally, the present discounted value of damages associated with a small unit increase in the emissions ofgas 

### i at time zero can be written as 

# I

[fJD(t)/fJE1(0)]e-"dt = 

# I[

# i [fJD(t)lfJR(r)][fJR(r)/fJE1(0)]dr] e-"dt 0 0 0 

# I[

# 1

**~(t,r)a,(r)dr] e~dt**  - 

# I a,(r) [

# I

**~(t,r)e-"dt]** dr (1)



# =

# Ia 1(r)o(r)dr.
0



### That is, the discounted damage caused by a unit increase in 

## ElOJ is equal to a weighted integral of

### alr}, where the weights do not depend on i. 

### This follows because emissions ofgreenhouse gases are (for the moment) assumed to cause only climate-related net damages, and such damages are only caused by changes in radiative forcing. The quantity o(r) 

### is the derivative ofthe present discounted value ofmarginal damages with respect to radiative forcing at time T.

### 252 I The Energy Journal
Note that in (7) the discount rate, 

### r, is applied to the dollar values of incremental damages over time. In this setting, as in general, the rationale for discounting of monetary values rests on some mixture of impatience and the productivity of investment, and the choice of an appropriate discount rate for publicpolicy analysis involves choosing the right mixture, as well as adjustment for risk and the effects of taxes.5 While a detailed analysis of this issue is beyond the scope ofthis essay, the importantpointis that the 

### economic problem ofchoosing the "correct" 

### r in (7) is well-defined, which it was not inthe context of (3). If o(r) in (7) were known for aliT, the argument so far implies that it should be used to replace the discount factor in (3) to compute indices ofglobal warming damage, and ratios of those indices could then be used to compare greenhouse gases. But this function is not known, importantly because great uncertainty surrounds the level ofdamages associated with any particularpattern ofclimatechange. Nonetheless, under some assumptions, the

### shape ofo(t) takes
on a familiar form, and uncertainties regarding the level ofthat function cancel out across gasses when comparisons are made in ratio form, as in (2) and (3). Sufficient conditions for this to be true are that following hold forj 

## = 1 , .

.., M: 

### f3p,v) = 

### f3Je 8'

### for t = 

### v, = 0

### for t 

### > v; riv,r) = 

### riv-r)for 

### v

**~** 1'. (8)

Setting the {31 = 0 for t 

### > v can be viewed as simply a convention on the measurement ofdamages. The assumption that the 

### f3/t,t) grow at rate g for all j and t is quite restrictive in principle but less so in practice, given the sketchy nature ofourknowledge ofthe likely costs ofglobal change. Nordhaus (1991, p. 925), for instance, argues in effect that 

### g should be set equal to the rate ofeconomic growth in "resource steady state" -when "all physical flows intheglobaleconomy are constanteventhough (because ofresource-augmenting technical change) the real value ofeconomic activity may be increasing." Out of such an equilibrium, one might argue that 

### g should be less than the rate of aggregate economic growth, since marginal physical damages to natural systems seem unlikely to grow as rapidly as the global economy, and the share of economic activity accounted for by agriculture and other climate-sensitive activities is secularly declining. On the other hand, the 

### value of damages to natural systems may rise quite rapidly because environmental amenities are luxury goods, and, depending on the baseline pace ofclimate change, marginal sensitivity to climate-related damages may rise more rapidly than damages

5. In general, see Lind (1984) and the literature he cites. Heal (1991) and Cline (1992) focus
on discounting issues that arise in the context ofclimate change.Comparing Greenhouse Gases I 253
themselves. At any rate, it is possible to have an intelligenteconomic argument about what value or values ofg best summarize available information on likely future changes in the marginal effects ofclimate change. The second line of(8) assumes local linearity ofthe important climatedetermining natural processes. In the absence ofdiscontinuouschanges or sharp nonlinearitieswithin the relevant range, this assumption should notbe seriously misleading. It is, of course, a modest generalization of the single linear differential equation that determines climate in the models ofNordhaus (1991) and others. Substituting from (8) into the definition ofo(7) given in (7), we obtain

oo 00 

## [N ] o(r) = 

# J 4>(t,r)e-''dt = 

# f

**~** f3J-yp-r)e8'

r r J I (9)

where 

### 8 

# =r-g, and 

### Ais a constant, independent of r. Thus a comparison of greenhousegases basedondiscountedclimate-relateddamages, usingwhatmight be called a relative damage index 

### (RDJ), reduces to a comparison ofdiscounted radiative forcing, with discount rate 8: 6

00 

# l[iW(t)/c3E1 (0)]e-''dt 

# l Ot1 (r)e -Brdr 

### RDI; 0 0 

### 1,

... 'N. 00 

# l [c3D(t)/c3E1{0)]e-''dt 0 

# l
a 1 (r)e-8'dr 0 (10)

As discussed above, the economic problem ofchoosing 

### rand g, and thus(}, is in principle well-defined. In contrast, simply writing down (3) provides no economic (or other) basis for selecting a particular discount rate.

6. This conclusion also follows from the steady-state optimal growth analysis of Nordhaus (1991).254 I The Energy Journal


### SOME COMPLICATIONS
WhiletheRDIdeveloped above is an improvementon the GWP and the DGWP, there are at least two reasons why itis unlikely to be an adequate basis for policy decisions. First, as Reilly (1991) notes, the assumption that the only external effects of greenhouse gas emissions are climate-related is both strong and crucial to the sort of analysis performed above. CFC emissions have important effects on stratospheric ozone, for instance, and plant growth may be sensitiveto atmospheric concentrations ofC02• Efficientcontrol strategies must take such effects into account ifthey surpass some threshold level ofeconomic importance. Since external effects that are notclimate-related do notoperate via changes inradiative forcing, thepresence ofsuch effects in general rules outthe sort ofcancellationthatproduced (10). Specifically, even underassumptions (8), discounted net damages in general simplify only to

00 

# I[oD(t)loE,(O)]e-"dt = 

### A

# Ici;(r)e-8

### 'dr + J'1, 0 0 

### i = 

### 1,

... , N,

where 'Yi is the discounted present value ofthe derivative ofnon-climate-related net damages with respect to emissions of gas i at time 0. In order to compare greenhouse gases for policy purposes in this case, both A and the 'Yi must be explicitly calculated.7 The second reason why (10) is unlikely to provide a sound basis for policy decisions is that an analytical framework that took full and explicit account ofthe uncertainties and irreversibilitiesthat are important inthe climate change context would likely imply a basically different approach to comparing greenhouse gases.8 Over time, scientific and economic research will likely reduceuncertaintiesregardingnaturalclimate-ratedprocesses, damagefunctions, and costs ofadaptation and abatement. Most climate-related policy actions that have been widely discussed, both those focused on abatement ofgreenhouse gas emissions and onadaptation to changed climates, wouldhave long-livedeffects, and changes 

### inemissions ofat least some greenhouse gases will have long-lived effects on radiative forcing. All else equal, a policy that puts primary near-term emphasis on relatively long-lived gases (C02 in particular) would seem to be attractive because it provides insurance against learning that climate change is a more

7. SeeReilly and Richards (forthcoming) for anilluminatingdevelopmentofthis pointand some interesting calculations ofdiscounted net damages. 8. Forexplorationsofthe policy implications oftwo aspects ofthe uncertainty that is pervasive in this context, see Heal (1984, 1991) and Hendricks (1991).

### Comparing Greenhouse Gases I 255


### serious problem than it now seems. This effect is necessarily absent from any analysis that neglects uncertainty. Of course, to go beyond this intuitive argument, or even to provide an adequate defense for it, would require a fullblown analysis ofuncertainty in this context. Such an analysis would be 

### quite valuable for a host ofreasons that go well beyond the issues considered in this essay.



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